Web Puzzles

"This," started the guide, "is the Buttons Room."

The Druggar of Bongo Ghango - the chief of a large country by the River Ghango - looked around. "It's a nice room, but where are the buttons? The only ones I see are, ahem, the ones on your shirt."
"Your Highness. The buttons, which could start a nuclear armageddon in a matter of seconds, are there, behind that panel," replied the guide, while pointing at a large panel at the end of the room.
"How can that be??? Why? Anybody - a madman, for example - could come here and press those terrible buttons?"

"Your Highness, it is very safe actually: for every button there is a slot, where a magnetic card must be inserted to activate the corresponding button. No card, no button. To launch the missiles, all buttons must be activated and pressed, and only a handful of people have the magnetic cards, and each card contains a different code from all the other ones."

"But it's the same thing! Any of these persons could go completely ballistic and start a nuclear war."
"In that case, the only dangerous man is the President of the United States, as he is the only person that holds all the codes, which would allow him to press all the buttons. The other people that hold some codes are the Vice-President of the United States, the President of the Senate, the Secretary of State, the Chief of the Armed Forces, and the Dean of Harvard University. Each of these gentlemen holds an incomplete set of magnetic cards, and the distribution of the codes is such that, if the President of the United States is not available, the entire set of buttons can be activated by the Vice-President, together with anyone of the other four men. If both the President and the Vice-President of the United States are unavailable, the buttons can still be activated by any three of the other four men. Therefore, to launch the missiles, it is needed either the President, or the Vice-President plus anyone of the other four men, or any three of the other four men."

"What if someone tried to press randomly many buttons, one after the other?" asked the Chief.
"Nothing would happen with the missiles, but the room would fill up with a narcotic gas, and an alarm would alert the guards and the CIA."

"So, how many buttons are there, and how are they distributed between the Vice-President and the other four personalities?"

And that's the question we'll ask the reader: what is the minimum number of buttons, and how are they distributed?

Solution